Category: Number Theory

Question Number 217071 by ArshadS last updated on 28/Feb/25 $$\mathrm{Find}\mathrm{all}\mathrm{two}-\mathrm{digit}\mathrm{numbers}\mathrm{that}\mathrm{are}\mathrm{equal}\mathrm{to}\mathrm{four}\mathrm{times}\mathrm{the}\mathrm{sum}$$$$\mathrm{of}\mathrm{their}\mathrm{digits}.\mathrm{Solve}\mathrm{this}\mathrm{using}\mathrm{at}\mathrm{least}\mathrm{two}\mathrm{different}\mathrm{methods}$$$$\mathrm{and}\mathrm{verify}\mathrm{your}\mathrm{answers}.$$ Answered by som(math1967) last updated on 28/Feb/25 $$\:{x}+\mathrm{10}{y}=\mathrm{4}\left({x}+{y}\right) \…

Question Number 217030 by ArshadS last updated on 27/Feb/25 $$\mathrm{Find}\mathrm{all}\mathrm{positive}\mathrm{integers}\mathrm{n}\mathrm{such}\mathrm{that}$$$$\mathrm{n}+1\mathrm{divides}{\mathrm{n}}^2+1$$ Answered by issac last updated on 27/Feb/25 $$\frac{{n}+\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\frac{{n}+\mathrm{1}}{\left(\mathrm{1}+{n}\boldsymbol{{i}}\right)\left(\mathrm{1}−{n}\boldsymbol{{i}}\right)} \…

Question Number 216995 by Rasheed.Sindhi last updated on 26/Feb/25 $$\mathrm{Find}\mathrm{all}\mathrm{prime}\mathrm{numbers}\mathrm{p}\mathrm{and}\mathrm{q}$$$$\mathrm{such}\mathrm{that}$$$${\mathrm{p}}^2-{\mathrm{q}}^2=2024$$ Answered by Marzuk last updated on 26/Feb/25…

Question Number 216911 by ArshadS last updated on 24/Feb/25 $$Findallpositiveinteger\mathrm{x},\mathrm{y}suchthat$$$${\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{xy}=169$$ Answered by A5T last updated on 25/Feb/25 $$\mathrm{WLOG},\:\mathrm{let}\:\mathrm{x}\geqslant\mathrm{y} \…

Question Number 216783 by ArshadS last updated on 20/Feb/25 $$Findallpositiveintegersnsuchthat$$$$n^2+7n+6isperfectsquare.$$ Answered by mehdee7396 last updated on 20/Feb/25 $${n}^{\mathrm{2}} +\mathrm{7}{n}+\mathrm{6}={k}^{\mathrm{2}} \…